INFORME DE GESTIÓN

in Description Logics

In this section, we propose the representation of CTL* logic semantics in terms of DL language family SHK semantics. As previously stated at the beginning of Section 4.4, we aim at using the instance checking reasoning to

CHAPTER 4. DESCRIPTION LOGIC-BASED MODEL CHECKING 48 perform the model checking process. Since we want to perform finite model checking (closed-world assumption) that enables negation-as-failure property [33], we require the usage of epistemic operator K that assumes the common domain and rigid term of the knowledge base.

We only consider the restricted CTL* formulas, which is defined as fol- lows:

g1 ::= p|>|¬p|(g1∧ g2)|(g1∨ g2)|

(g1Ug2)|X g1|G g1|F g1 (4.16)

f1 ::= E [g1]. (4.17)

Figure 4.5: Syntax rules of the restricted CTL* formula

It should be noted that the quantifier (E ), temporal (U, F, G, X) and boolean (¬, ∧, ∨) operators are the same with the operators of the CTL* formulas. The difference lies in the path quantifier and the negation. We only allow the existential path quantifier (E ) and the occurrence of negation immediately before an atomic proposition.

Recall the definitions of p, g1, g2, s, π and KB, x from Section 4.2.2 and the first part of Section 4.4, respectively. Let Di and Daux and be the con- cept representing the sub formula gi and the additional auxiliary concept, respectively. The symbol σ denotes a sequence of Abox individuals, which is the description logics representation of trace π. Thus, σi denotes the Abox individual at the index i.

We propose the translation of the CTL* logic semantics to SHK seman- tics that are defined in Figure 4.6. The complete proofs of this translation can be found in Appendix A. The left hand of the equations are the se- mantics of CTL* and the right hand of the equations are the corresponding translations, which are represented in description logic language semantics.

The purpose of the epistemic operator, which is used to represent CTL* semantics in terms of description logic semantics, is to enable the closed- world reasoning. Closed-world reasoning facilitates the evaluation of failure as negation. To understand the meaning of the failure as negation, let us consider the formal semantics of CTL* showed in Figure 4.3. Based on CTL* semantics, we conclude that s 2 p is equivalent to s  ¬p. This means that the failure of the entailment is equivalent to the negation.

In description logic representation, we define

s 2 p (4.28)

s  pi ⇔ KB  KVi(x) (4.18) s  ¬pi ⇔ KB  ¬KVi(x) (4.19) s  E (g1) ⇔ KB  D1(x) (4.20) π  g1 ⇔ KB  D1(σ0) (4.21) π  g1∨ g2 ⇔ KB  (D1t D2)(σ0) (4.22) π  g1∧ g2 ⇔ KB  (D1u D2)(σ0) (4.23) π  X g1 ⇔ KB  (∃Knext.D1)(σ0) (4.24)

π  G g1 ⇔ hTKB∪ {Daux ≡ D1u ∃Knext.Daux}, AKBi

 Daux (σ0) (4.25)

π  F g1 ⇔ KB  (D1t ∃future.D1) (σ0) (4.26)

π  g1Ug2 ⇔ hTKB∪ {Daux ≡ D2t (D1u ∃Knext.Dauxu ∃future.D2)}

, AKBi  Daux (σ0) (4.27)

Figure 4.6: Formal semantics of a subset of CTL* logic in SHK as

KB 2 V (x) (4.30)

KB  ¬V (x), (4.31)

respectively. The individual x represents the state s and the concept V represents the set of individuals having the label p. Equation 4.30 states that it could not be deduced from the knowledge base that x is a member of V . Equation 4.31 states that the individual x is not a member of the concept V . In the CTL* semantics, both equations are equivalent.

However, in description logic semantics, Equation 4.30 and Equation 4.31 are not equivalent, although both equations represent Equation 4.28 and Equation 4.29, respectively. This is due to the fact that description logic semantics have the open-world reasoning property. This means that if the knowledge base cannot deduce that x is a member of V , it does not necessarily imply that x is not a member of V . Since the knowledge base does not have sufficient knowledge to answer the query, it cannot deduce the fact in answering the query.

Therefore, we introduce the epistemic operator K in order to have the closed-world reasoning property in the description logic reasoning. The epis- temic operator K immediately appears in the atomic proposition and the

CHAPTER 4. DESCRIPTION LOGIC-BASED MODEL CHECKING 50 state transition. (see equations 4.18, 4.19, 4.24 - 4.25 and 4.27.)

Expressing the semantics of modal operators X and F in description logic is very simple, since we use next relationship and transitive relationship future ∈ R+ to build the semantics. It should be noted that the relationship next is a sub-relationship of future. Thus, we define the transitivity of future and the assertion next v future in the Tbox.

To express the semantics of modal operator G, we use the cyclic con- cept definition for expressing the models of infinite traces that satisfy the G operator. Since the Tbox contains cyclic concept definitions, we require a particular interpretation semantics, which are used in the reasoning engine.

It should be noted that the description logic language formalism has three different types of interpretation semantics [68]. They are: least-fixpoint se- mantics, greatest fixpoint semantics and descriptive semantics. The least (greatest) fixpoint semantics consider the smallest (largest) set of models with cyclic relationships, which satisfy cyclic concept definitions. The de- scriptive semantics, however, do not consider the models with cyclic rela- tionships [7]. Thus, the descriptive semantics can not be used to represent the semantics of G modal operator, which is only satisfied by a cyclic model containing some infinite traces.

If the Tbox does not contain any cyclic concept definition, then these interpretation semantics are all the same. Thus, the choice of the interpre- tation semantics used in the reasoning engine will not affect the reasoning result [96]. The descriptive semantics are usually used in most reasoning engines, due to their simplicity of implementation.

However, if the Tbox contains some cyclic concept definitions, then these three interpretation semantics with regard to the cyclic concepts differ from each other [96]. Baader points out that the least-fixpoint semantics always consider a set of empty models, which satisfy the cyclic concepts’ interpreta- tion [6]. Such interpretation semantics are uninteresting to us. On the other hand, the greatest-fixpoint semantics can interpret cyclic concept definitions as the largest set of models, containing the cyclic relationships. This suits our need to correctly interpret the G modal operator. Therefore, we require that the reasoning engine should support the greatest-fixpoint semantics in order to correctly represent the semantic of G modal operator.

Although the translation of the semantic ofU operator uses cyclic concept definition, we can use descriptive semantics to correctly capture the meaning of U operator. This is due to the fact that U operator can be satisfied by finite traces.

Soundness and Correctness of the CTL* Logic. The most important properties of any logic are the soundness and completeness properties. In a nutshell, these properties assure that the truth values of the derivation of any formula, which is either derived using natural deduction (`) or interpretation model (), are equivalent. In [60] soundness and correctness are defined as follows:

Definition 23 (Soundness) A formal system F S is sound, if and only if for each set of formulas Z and every formula A:

Z ` A ⇒ Z  A.

This definition says that every A, which is derivable from Z using the deduc- tion rules of the corresponding logic, is also satisfied by the interpretation models, which satisfy all formulas in Z.

Definition 24 (Completeness) A formal system F S is complete, if and only if, for each set of formulas Z and every formula A:

Z  A ⇒ Z ` A.

Conversely to the Definition 23, this definition says that every A, which is satisfied by the interpretation models satisfying all formulas in Z, is derivable from Z by using the deduction rules of the corresponding logic.

The original interpretation model of restricted CTL* is namely Kripke model. In this thesis, we represent the Kripke model in terms of Description Logic language and use the Description Logic inference mechanism to perform the logical entailment () of the formal semantics of restricted CTL*. In other words, we try to perform the entailment of restricted CTL*, which is based on Kripke model, by using the Description Logic representation and its inference mechanism.

As we can see in Figure 4.6, each semantic entailment of restricted CTL* is semantically equivalent to the corresponding concept represented in De- scription Logic according to the inference mechanism (instance checking). Therefore, we say that we emulate the formal semantics of restricted CTL* by using Description Logic representation and its inference mechanism. This implies that our Description Logic-based representation of the semantics of restricted CTL* also preserve the soundness and the correctness of restricted CTL*.

CHAPTER 4. DESCRIPTION LOGIC-BASED MODEL CHECKING 52